Certified Reduced Basis Method for the Electric Field Integral Equation

• Published in: SIAM journal on scientific computing 2012-01, Vol.34 (3), p.A1777-A1799
• Main Authors: Hesthaven, J S, Stamm, B, Zhang, S
• Format: Article
• Language: English
• Subjects: Applied mathematics; Approximation; Boundary element method; Electric fields; Error analysis; Errors; Estimators; Integral equations; Mathematical analysis; Mathematical models; Mathematics; Norms; Numerical Analysis
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/110848268

In [B. Fares et al., J. Comput. Phys., 230 (2011), pp. 5532--5555], a reduced basis method (RBM) for the electric field integral equation (EFIE) using the boundary element method (BEM) is developed, based on a simplified a posteriori error estimator for the greedy-based snapshot selection. In this paper, we extend this work and propose a certified RBM for the EFIE based on a mathematically rigorous a posteriori estimator. A central difficulty of the certified method is that the intrinsic solution space of the EFIE is ${\boldsymbol{H}^{-1/2}_{{\rm div}}(\Gamma)}$, inducing a relatively complicated norm. Since the measured error consists of the difference between the reduced basis solution and the boundary element solution, which is a member of the discrete boundary element space, we clarify that the intrinsic norm can be replaced by an alternative norm and in this work use the $\boldsymbol{H}({\rm div})$-norm, which is computable and demonstrated to not degrade the quality of the error estimator. A successive constraint method (SCM) for complex matrices is discussed in detail, and numerical tests for the SCM and then the certified RBM confirm the analysis. [PUBLICATION ABSTRACT]